Noticias en español
Symmetric positive solutions to nonlinear Choquard equations with potentials.
Abstract: I will present some existence results for a class of Choquard equations in which the potential has a positive limit at infinity and satisfies suitable decay assumptions. Also, it is taken invariant under the action of a closed subgroup of linear isometries of RN. As a consequence, the positive solution found is invariant under the same action. I will mainly focus on the physical case involving a quadratic nonlinearity. Joint work together with Liliane Maia and Benedetta Pellacci.
Read MoreInfinitely many entire solutions to a mixed dispersion Schrödinger equation with generic non-linearity.
Abstract. I will present a multiplicity result for the mixed dispersion non-linear Schrödinger equation \[\Delta^2u−\beta \Delta u=g(u), \qquad \mbox{in}\quad\mathbb{R}^N \] focusing on the case $N \geq 5$, where the non-linearity $g\colon\mathbb R\to \mathbb R$ satisfies assumptions in the spirit of Berestycki & Lions.After showing some compactness results, I will demonstrate how the variational approach of [1], which makes use of auxiliary functionals, can be used for this problem.
Read MoreOn the singular Q-curvature problem.
Abstract. The connections between geometry and partial differential equations have been extensively studied in the last decades. In particular, some problems arising in conformal geometry, such as the classical Yamabe problem, can be reduced to the study of PDEs with critical exponent on manifolds. More recently, the so-called Q-curvature equation, a fourth-order elliptic PDE with critical exponent, is another class of conformal equations that has drawn considerable attention by its relation with a natural concept of curvature. In this talk, I would like to discuss how fixed point methods...
Read MoreFully nonlinear models with non-homogeneous degeneracy and related problems.
Abstract. In this talk we establish sharp C1,β regularity estimates for bounded solutions loc of a class of fully nonlinear elliptic PDEs with non-homogeneous degeneracy, whose model equation is given by [|Du|p +a(x)|Du|q]M+λ,Λ(D2u)=f(x,u) in Ω, for a bounded domain Ω ⊂ RN, and suitable data p,q ∈ (0,∞), a and f. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. Finally, we present some connections of our findings with a variety of nonlinear free boundary problems in the theory of elliptic...
Read MoreA nonlocal version of the inverse problem of Donsker and Varadhan.
Abstract: In their seminal paper of 1976, M.D. Donsker and S.R.S. Varadhan addressed the following “inverse problem”: let consider two linear, second-order, uniformly elliptic operators L1, L2 with the form Liφ = Div(Ai(x)Dφ) + bi(x) · Dφ, i = 1, 2. If for every domain Ω and every smooth potential V , the operators L1 + V and L2 + V have the same principal eigenvalue in Ω, then the diffusions are equal (A1 = A2), and either L1φ = L2(uφ)/u for some L2-harmonic function u, or L1φ = L∗2(uφ)/u for some L∗2-harmonic function u. In this talk we report a nonlocal a version of this problem, where...
Read MoreSome ideas to obtain energy in Chile.
Abstract: In this talk we will present advances of a program that studies the modeling, analysis and control of models that describe the dynamics of wave energy converters (WEC) devices. In particular we will study two nonlinear models of an oscillating water column, where the one-dimensional shallow water equations in the presence of this device are essentially reformulated as two transmission problems: the first one is associated with a step in front of the device and the second one is related to the interaction between waves and a fixed partially-immersed structure. We will close the...
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